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Journal of Physics A Mathematical and General

Breakdown of dynamic scaling at the percolation threshold

S. Jain

Published Feb 1, 1986• 23 citations
Publisher

Abstract

The critical dynamics of the two-dimensional Ising model at the bond percolation threshold is investigated by Monte Carlo simulations on a 64*64 lattice. Conventional dynamic scaling breaks down at low temperatures: the logarithm of the relaxation time depends quadratically upon the logarithm of the thermal correlation length. The coefficients of the quadratic and linear terms are 0.51 and 3.25, respectively. The results are compared with recent experimental and analytic work.

DOI
10.1088/0305-4470/19/2/007

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The critical dynamics of the two-dimensional Ising model at the bond percolation threshold is investigated by Monte Carlo simulations on a 64*64 lattice. Conventional dynamic scaling breaks down at low temperatures: the logarithm of the relaxation time depends quadratically upon the logarithm of the thermal correlation length. The coefficients of the quadratic and linear terms are 0.51 and 3.25, respectively. The results are compared with recent experimental and analytic work.
The authors use real-space renormalisation group techniques to study the kinetic Ising model (with Glauber dynamics) on bond-diluted d-dimensional hypercubic lattices, at the percolation threshold. The results confirm and generalise the recently proposed breakdown of dynamic scaling and they predict a dependence on dimensionality of the factors A and B on the temperature-dependent dynamic exponent z=A log xi +B (where xi is the thermal correlation length). These factors also depend on the lattice scaling factor (b); the authors suggest that the limit b=1 may be taken as the best choice for the important coefficient A, leading to A=d(d-1)/2 nu p .
The critical dynamics of the diluted two-dimensional Ising model below the percolation threshold is studied. The dynamics exhibits a crossover from a concentration-dependent behaviour for xi T >> xi p to a percolating lattice type behaviour for xi T << xi p , where xi T and xi p are the thermal and percolation correlation lengths, respectively. As a consequence, the dynamical exponents A and B are shown to be the limits of dynamical functions A(x) and B(x), respectively; here, x=ln xi p /ln xi T . Extensive Monte Carlo simulations support our predictions.
The relaxation time \ensuremath{\tau} and the thermal correlation length ${\ensuremath{\xi}}_{\mathit{T}}$ in randomly diluted Ising magnets near the percolation threshold are related by the singular dynamic scaling ln\ensuremath{\tau}=f(ln${\ensuremath{\xi}}_{\mathit{T}}$), where f(x)=Ax2{\mathit{Ax}}^{2}Ax2+Bx+C, with constants A,B,C. We investigate the dimensionality dependence of A by Monte Carlo simulation and compare our observations with the theoretical predictions.